Restricted range simultaneous approximation and interpolation with preservation of the norm Annales

Let (F. ; ~ ~ ) be a complete non-archimedean non-trivially valued division ring. w ith valuation ring V. Let X be a compact 0-dimensional Hausdorff space. and let D(X) be the ring of all continuous functions f from X into V equipped with the supremum norm. Let A C D(X). Assume that for every ordered pair (s. t) of distinct elements of X. there is some multiplier of A; say ~;. such that = 1 and = 0. Assume that A contains the constants. We show that A is uniformly dense in D(X ). and when A is an interpolating family then simultaneous approximation and interpolation, with preservation of the norm, by elements of .4 is always possible. We apply this to the case of von Neumann subsets and to the case of restricted range polynomial algebras. 1991 Mathematics subject classification: 46S10.


Introduction
Throughout this paper X is a compact Hausdorff space which is 0-dimensional i.E. , for any point ir and any open set 44 containing ~. there exists a closed and open set ~l~' with ir E C .A. and (.F. j -j) is a complete. non-Archimedean non-trivially valued division ring.
We denote by V the valuation ring of F. i.e., V = {t e F: |t| I 1}, and by D(X) the set of all continuous functions from the space fY into Y. equipped with the topology of uniform convergence on X. determined by the metric d defined by = ~f -g~ = sup{|f(x) -g(x)|:x E X} for every pair, f and g, of elements of D(X).
Our aim is to use the idea of T.J. Ransford (see ~7~); to prove results in D(X) that are analogous to those in C(X; [0. and C(X; F), which were proved in (5] and [6]. respectively. To avoid trivialities we assume that A' has at least two points. ' Definition 1 A non-empty subset A C D(X) is said to be a von Neumann subset if 03C8 , + (1 -03C6)~ belongs to A, whenever 03C6, 03C8 and ~ belong to A . . Clearly, if A C D(X) is a von Neumann subset containing the constant functions 0 and 1. then the follow ing properties are true: (i) if 'f E A. then belongs to A: (ii) if ; and ~,~ belong to .4. then ~;y belongs to A.
When A C D(X) has properties (i) and (ii), we say that A has property y'. This definition is motivated by the similar one introduced by R. I. Jewett, who in [1] proved the variation of the Veierstrass -Stone Theorem stated by von Neumann in [8]. Definition 3 A subset A C D(X) is said to be strongly separating over X. if given any ordered pair (x. y) E .Y x X. with y; there exists a function.; E A such that ;(x) = 1 and ~( r~) = 0. Lemma 1 Let AI C D(X) ) be a subset which has proper V and is strongly separating over X. . Let be a clopen proper subset of X . For each 03B4 > 0 , there -is 03C6 E M such that~( t) -lj b, for all t E N. (I) Proof. This result is essentially Lemma 1 of Prolla [6]. For the sake of completeness we include here its proof. Fix y E X. y ~ N. Since ~I is strongly separating; for each t E N, there is ;t E M such that 03C6t(y) = 1. = 0. By continuity there is a neighborhood V(t)  (2)  Clearly .~' ~ 0. because X E .F. Let us order .~' by set inclusion. Let C be a totally ordered non-empty subset of 0.
Let S = n{T: T E C}. Clearly. S is closed. If J is a finite subset of C. there is some To E .J such that T0 ~ T for all T E J. Hence To = n{T; T E J}. Now 0 and by compactness 5' 7~ 0. Hence S E ~'(fY). We claim that S E -F. Clearly, dist ( fs~: Js) d. Suppose that dist ( fs; As) d and choose a real number r such that dist (fS: AS) r d. By definition of dist ( f 5: As) there exists g E A such that r for all x E S. Let Then U is open and contains S. By compactness, there is finite subset .J C C such that n{T: T E J} C L'. Let To E J be such that To C T for all T E .J. Then n{T: T E J} = To and so To C L'. Hence r for all t E To. and so dist (fT0); AT4 r d, which contradicts the fact that To E F. This contradiction establishes our claim that dist (is; As) ) = d. Therefore S is a lower bound for C in:F. By Zorn's Lemma there exists a minimal element in 7. and this element satisfies all our requirements. a 2. The Main Results Theorem 1 Let A C D(X) be a non-empty subset. whose set of multipliers is strongly separating over JB". . For each f E D(X ), , there is some x E X such that (2) F., for all t ~ N.
The function k = jg + ( 1 -)h belongs to A. 'Ve claim that k(t)~ r for all t E S. Let t E S. There are two cases to consider, namely t e Y and t E Z. Therefore k(t)~ r, for all t E S and dist ( fs, As) r d, a contradiction.
Remark. If ,4 c D(X } is as in Theorem i and A(x) ~ {0, I }, for every x E .Y, then it follows that the closure of .~ contains the characteristic function of each clopen subset of X . Indeed, let S c .Y be a clopen subset of X and let f be its characteristic function. Let x E X be given by Theorem 1. Now f(x) is either 0 or 1 and therefore A(x} contains f(x) and so dist ( f , A) = 0. Corollary 1 Let A c D(X} be a von Neumann subset which is strongly separating over X. . For each f E D{X ). . there is some ~ E X such that be the set of all multipliers of A. Since A is a von Neumann subset, we see that A C M. Hence M is strongly separating too, and the result follows from Theorem 1. t7 Theorem 2 Let :-1 c D(X) ) be a non-empty subset. whose set of multipliers is strongly separating over .Y. . Let f E D(X } and ~ > 0 be given. The following are equivalent: (1) there is some g E A such that ~f -g~ ~, (2) for each t E X,there is some gc E A such that ! f (t) -~.
Proof. Clearly. (1) ~ (2). Conversely, assume that (2) holds. Let x E X be given by Theorem  By (2) applied to t = x, there is some gx E A such that gx(x)| ~. Hence dist { f (x); A(x)) ~. By (*) above. dist { f A) ~, and therefore some g E A such that~ f -g~~ ~ can be found. Hence (1) is valid. O Corollary 2 Let A C D(X} ) be a von Neumann subset which is strongly separating over X . Let f E D(X } and ~ > 0 be given. The following are equivalent: (1) there is some g E A such that ~f -g~ ~, Then A .is uniformly dense in D(X).
Proof. Let f E D(X ). By Theorem 1. there is some x E X such that dist ( f : :-I) = dist ( f (x}; A(x}). Corollary 3 Let..4 C D(X ) be a von Neumann subset which is strongly separating over X. and for each a E L' and x E X there is 03C6 E A such that = a. Then A is uniformly dense in D(X). Corollary 4 Let be a subring of D(..Y) which is strongly separating over X and W(x) = V. for each x E ~'. Then is uniformly dense in D(X ).
Proof. Clearly, every subring of D(X) is a von Neumann subset. Q Remark. The valuation ring V is a topological ring with unit, and has a fundamental system of neighborhoods of 0 which are ideals in V. Hence Theorem 32 of Kaplansky [2] applies, giving an alternate proof for Corollary 4.

Examples
Let us give some examples of von Neumann subsets of D(X) which are strongly separating over X. Let us first remark that a separating subring of D(X) is not necessarily strongly separating over X. . The set = { f E D(X ); |f(x)| 1, for all x E X} is an example of a separating subring of D(X) infact, it is a closed two-sided ideal of D(X ), which is not strongly separating. Indeed no function in W can take the value 1 at any point in X. Further examples can be found. Indeed, for a fixed point E ,Y let us define Wa = { f E D(X); f(a) = 0}.
Clearly, Wa is a subring of D(X). Now Wa is separating over X. Indeed, let x i= y be given in X. If x = a or y = a, the function.; E D(X) which is zero at a and one at the other point is such that 03C6(x) ~ 03C6(y) and 03C6 E Wa. In case x ~ a and y ~ a,let 03C6 E D(X) be such that cp(a) = 0 and = 1, and let E D(X) be such that = 0 and 03C8(y) = 1.
Then 11 = E it] and = 0 while = 1. On the other hand, Wa is not strongly separating over X. . For every ordered pair (a. x). with x. there is no function °° E yi~u such that 03C6(a) = 1 and ;(x) = 0. Indeed. 03C6 E Wa implies f'(a) = 0, and so Wa is not strongly separating over X.
Example 1 The collection A of the characteristic functions of all the clopen subsets of X is a von Neumann subset of D(X}, containing 0 and 1. and moreover. since X is a o-dimensional compact Hausdorff space. A is strongly separating over X. 1~. where is the p-adic field. Then the unitary subalgebra i~' of all polynomials q : Qp -; Qp is separating over X. . By Proposition 1, Prolla [6], A = {q E W; q(X) C V} is strongly separating over X. . Clearly. A is a von Neumann subset containing the constants in D(..Y). Example 4 Let be a finite partition of X into clopen subsets, i.e., the set I of indices is finite, each Si is a clopen set, Si n Sj = 0 for all i and X = ~i~ISi. For each i E I. let ~;t be the characteristic function of Si and let a~ E V. Consider the function ~ E D(X) defined bv 03C6(x) -03A3 03BBi03C6i(x) iel for all x E X. Let .4 C D(X) be the collection of all functions ;~ defined as above. Then .4 satisfies all the hypothesis of Theorem 3 and therefore is uniformly dense in D(X). Definition 4 A non-empty subset A C D(X) is said to be a restricted range polynomial algebra if for every choice 03C61,...,03C6n E A and q : F a polynomial in n-variables such that ~,2(x~..... 1 for all x E X. the mapping x -~ belongs to A. Notice that the polynomials (u1, u2} -u1 + u2. (u1, u2) -u1u2 and (u1, u2) ~ u1 -u2 are such that V x V is mapped into V, and therefore any restricted range polynomial algebra is a subring of and a fortiori a von Neumann subset. Notice that any restricted range polynomial algebra contains all the constant functions with values in V. Proposition 1 Let.4 C D(X) be a restricted range polynomial alyebra which is separating over X. . Then A is strongly separating over X , Proof. Let (8, t) be an ordered pair of distinct elements of X. By hypothesis, there exists ~ E A such that ~(t).
Proof. By Proposition 1. A is strongly separating. On the other hand A contains all the constant functions with values in V. Hence A(x) = V, for every x E .Y. Since A is a von Neumann set. the result follows from Corollary 3.Or else, notice that A is a subring and then apply Corollary 4.

Simultaneous Aproximation and Interpolation
Definition 5 A non-empty subset A c D(X) is called an interpolating family for D(X) if, for every f E D(X) and every finite subset S C X. there exists 9 E A such that y(x) = I(x) for ' Theorem 4 Let C D(X) be is an interpolating family for D(X ), whose set of multipliers is strongly separating over .Y. Then, for every f E D(.Y) every ~ > 0 and every finite set S C X, there exists 9 E A such that ~f -g~ ~, ~g~ = and g(t) = for all t E S.
Proof. Let A = {g E g(t) = J(t) for all t E S}. Since is an interpolating family for D(X), the set A is non-empty. It is easy to see that every multiplier of L4' is also a multiplier of A. Hence the set of multipliers of A is strongly separating over X. Consider the point it E X given by Theorem  Corollary 6 Let II' c D(X) be an interpolating family for D(X) wh~ich is a von Neumann subset and which is strongly separating over X . Then. for every f E D(X), every ~ > 0 and every finite set S C X , there exists g E W such that ~f -g~ ~, ~g~ = ~f~, and g(t) = f(t) for all t ~ S.
Proof. The set W is contained in the set M of its multipliers and Corollary 6 follows from Theorem 4.

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Remark. If c D(X) is an interpolating family for D(X) which is strongly separating over X and which is a subring of D(X }, then Corollary 6 applies to it. Corollary 7 Let W C D(X) be an interpolating family for D(X } which is a restricted range polynomial algebra and which is separating over X . Then. for every f E D(X ). , every ~ > o, and every finite set S C X, there exists g E such that -g~ ~, ~g~ = and g(t) = f(t) fo.r all t E S.
Proof. We know that every restricted range polynomial algebra is a von Neumann subset. By Proposition 1.
is strongly separating. The result now follows from the previous Corollary.